Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 76.7% → 96.7%

Time: 20.5s

Alternatives: 10

Speedup: 1.0×

• Unsound rule application detected in e-graph. Results may not be sound.

Specification

Math

\[\begin{array}{l} \\ \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (*
  (cos (- (/ (* K (+ m n)) 2.0) M))
  (exp (- (- (pow (- (/ (+ m n) 2.0) M) 2.0)) (- l (fabs (- m n)))))))

C

double code(double K, double m, double n, double M, double l) {
	return cos((((K * (m + n)) / 2.0) - M)) * exp((-pow((((m + n) / 2.0) - M), 2.0) - (l - fabs((m - n)))));
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    code = cos((((k * (m + n)) / 2.0d0) - m_1)) * exp((-((((m + n) / 2.0d0) - m_1) ** 2.0d0) - (l - abs((m - n)))))
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	return Math.cos((((K * (m + n)) / 2.0) - M)) * Math.exp((-Math.pow((((m + n) / 2.0) - M), 2.0) - (l - Math.abs((m - n)))));
}

Python

def code(K, m, n, M, l):
	return math.cos((((K * (m + n)) / 2.0) - M)) * math.exp((-math.pow((((m + n) / 2.0) - M), 2.0) - (l - math.fabs((m - n)))))

Julia

function code(K, m, n, M, l)
	return Float64(cos(Float64(Float64(Float64(K * Float64(m + n)) / 2.0) - M)) * exp(Float64(Float64(-(Float64(Float64(Float64(m + n) / 2.0) - M) ^ 2.0)) - Float64(l - abs(Float64(m - n))))))
end

MATLAB

function tmp = code(K, m, n, M, l)
	tmp = cos((((K * (m + n)) / 2.0) - M)) * exp((-((((m + n) / 2.0) - M) ^ 2.0) - (l - abs((m - n)))));
end

Wolfram

code[K_, m_, n_, M_, l_] := N[(N[Cos[N[(N[(N[(K * N[(m + n), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision] * N[Exp[N[((-N[Power[N[(N[(N[(m + n), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision], 2.0], $MachinePrecision]) - N[(l - N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Initial Program: 76.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (*
  (cos (- (/ (* K (+ m n)) 2.0) M))
  (exp (- (- (pow (- (/ (+ m n) 2.0) M) 2.0)) (- l (fabs (- m n)))))))

C

double code(double K, double m, double n, double M, double l) {
	return cos((((K * (m + n)) / 2.0) - M)) * exp((-pow((((m + n) / 2.0) - M), 2.0) - (l - fabs((m - n)))));
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    code = cos((((k * (m + n)) / 2.0d0) - m_1)) * exp((-((((m + n) / 2.0d0) - m_1) ** 2.0d0) - (l - abs((m - n)))))
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	return Math.cos((((K * (m + n)) / 2.0) - M)) * Math.exp((-Math.pow((((m + n) / 2.0) - M), 2.0) - (l - Math.abs((m - n)))));
}

Python

def code(K, m, n, M, l):
	return math.cos((((K * (m + n)) / 2.0) - M)) * math.exp((-math.pow((((m + n) / 2.0) - M), 2.0) - (l - math.fabs((m - n)))))

Julia

function code(K, m, n, M, l)
	return Float64(cos(Float64(Float64(Float64(K * Float64(m + n)) / 2.0) - M)) * exp(Float64(Float64(-(Float64(Float64(Float64(m + n) / 2.0) - M) ^ 2.0)) - Float64(l - abs(Float64(m - n))))))
end

MATLAB

function tmp = code(K, m, n, M, l)
	tmp = cos((((K * (m + n)) / 2.0) - M)) * exp((-((((m + n) / 2.0) - M) ^ 2.0) - (l - abs((m - n)))));
end

Wolfram

code[K_, m_, n_, M_, l_] := N[(N[Cos[N[(N[(N[(K * N[(m + n), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision] * N[Exp[N[((-N[Power[N[(N[(N[(m + n), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision], 2.0], $MachinePrecision]) - N[(l - N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Alternative 1: 96.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \cos M \cdot e^{\left|m - n\right| - \left({\left(\frac{m + n}{2} - M\right)}^{2} + \ell\right)} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (* (cos M) (exp (- (fabs (- m n)) (+ (pow (- (/ (+ m n) 2.0) M) 2.0) l)))))

C

double code(double K, double m, double n, double M, double l) {
	return cos(M) * exp((fabs((m - n)) - (pow((((m + n) / 2.0) - M), 2.0) + l)));
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    code = cos(m_1) * exp((abs((m - n)) - (((((m + n) / 2.0d0) - m_1) ** 2.0d0) + l)))
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	return Math.cos(M) * Math.exp((Math.abs((m - n)) - (Math.pow((((m + n) / 2.0) - M), 2.0) + l)));
}

Python

def code(K, m, n, M, l):
	return math.cos(M) * math.exp((math.fabs((m - n)) - (math.pow((((m + n) / 2.0) - M), 2.0) + l)))

Julia

function code(K, m, n, M, l)
	return Float64(cos(M) * exp(Float64(abs(Float64(m - n)) - Float64((Float64(Float64(Float64(m + n) / 2.0) - M) ^ 2.0) + l))))
end

MATLAB

function tmp = code(K, m, n, M, l)
	tmp = cos(M) * exp((abs((m - n)) - (((((m + n) / 2.0) - M) ^ 2.0) + l)));
end

Wolfram

code[K_, m_, n_, M_, l_] := N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision] - N[(N[Power[N[(N[(N[(m + n), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision], 2.0], $MachinePrecision] + l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 74.8%

   \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
2. Step-by-step derivation

   1. associate-/l* 74.8%

      \[\leadsto \cos \left(\color{blue}{\frac{K}{\frac{2}{m + n}}} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]

   2. associate--r- 74.8%

      \[\leadsto \cos \left(\frac{K}{\frac{2}{m + n}} - M\right) \cdot e^{\color{blue}{\left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell\right) + \left|m - n\right|}} \]

3. Simplified 74.8%

   \[\leadsto \color{blue}{\cos \left(\frac{K}{\frac{2}{m + n}} - M\right) \cdot e^{\left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell\right) + \left|m - n\right|}} \]

4. Taylor expanded in K around 0 97.7%

   \[\leadsto \color{blue}{\cos \left(-M\right)} \cdot e^{\left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell\right) + \left|m - n\right|} \]
5. Step-by-step derivation

   1. cos-neg 97.7%

      \[\leadsto \color{blue}{\cos M} \cdot e^{\left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell\right) + \left|m - n\right|} \]

6. Simplified 97.7%

   \[\leadsto \color{blue}{\cos M} \cdot e^{\left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell\right) + \left|m - n\right|} \]

7. Final simplification 97.7%

   \[\leadsto \cos M \cdot e^{\left|m - n\right| - \left({\left(\frac{m + n}{2} - M\right)}^{2} + \ell\right)} \]

Alternative 2: 68.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -60000:\\ \;\;\;\;\cos M \cdot e^{m \cdot \left(m \cdot -0.25\right)}\\ \mathbf{elif}\;m \leq 5.5 \cdot 10^{-238}:\\ \;\;\;\;\cos M \cdot e^{\left|m - n\right| - \left(\ell + M \cdot M\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{n \cdot \left(n \cdot -0.25\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= m -60000.0)
   (* (cos M) (exp (* m (* m -0.25))))
   (if (<= m 5.5e-238)
     (* (cos M) (exp (- (fabs (- m n)) (+ l (* M M)))))
     (* (cos M) (exp (* n (* n -0.25)))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -60000.0) {
		tmp = cos(M) * exp((m * (m * -0.25)));
	} else if (m <= 5.5e-238) {
		tmp = cos(M) * exp((fabs((m - n)) - (l + (M * M))));
	} else {
		tmp = cos(M) * exp((n * (n * -0.25)));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (m <= (-60000.0d0)) then
        tmp = cos(m_1) * exp((m * (m * (-0.25d0))))
    else if (m <= 5.5d-238) then
        tmp = cos(m_1) * exp((abs((m - n)) - (l + (m_1 * m_1))))
    else
        tmp = cos(m_1) * exp((n * (n * (-0.25d0))))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -60000.0) {
		tmp = Math.cos(M) * Math.exp((m * (m * -0.25)));
	} else if (m <= 5.5e-238) {
		tmp = Math.cos(M) * Math.exp((Math.abs((m - n)) - (l + (M * M))));
	} else {
		tmp = Math.cos(M) * Math.exp((n * (n * -0.25)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if m <= -60000.0:
		tmp = math.cos(M) * math.exp((m * (m * -0.25)))
	elif m <= 5.5e-238:
		tmp = math.cos(M) * math.exp((math.fabs((m - n)) - (l + (M * M))))
	else:
		tmp = math.cos(M) * math.exp((n * (n * -0.25)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (m <= -60000.0)
		tmp = Float64(cos(M) * exp(Float64(m * Float64(m * -0.25))));
	elseif (m <= 5.5e-238)
		tmp = Float64(cos(M) * exp(Float64(abs(Float64(m - n)) - Float64(l + Float64(M * M)))));
	else
		tmp = Float64(cos(M) * exp(Float64(n * Float64(n * -0.25))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (m <= -60000.0)
		tmp = cos(M) * exp((m * (m * -0.25)));
	elseif (m <= 5.5e-238)
		tmp = cos(M) * exp((abs((m - n)) - (l + (M * M))));
	else
		tmp = cos(M) * exp((n * (n * -0.25)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[m, -60000.0], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(m * N[(m * -0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 5.5e-238], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision] - N[(l + N[(M * M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(n * N[(n * -0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -6e4

   1. Initial program 74.6%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 74.6%

         \[\leadsto \cos \left(\color{blue}{\frac{K}{\frac{2}{m + n}}} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]

      2. associate--r- 74.6%

         \[\leadsto \cos \left(\frac{K}{\frac{2}{m + n}} - M\right) \cdot e^{\color{blue}{\left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell\right) + \left|m - n\right|}} \]

   3. Simplified 74.6%

      \[\leadsto \color{blue}{\cos \left(\frac{K}{\frac{2}{m + n}} - M\right) \cdot e^{\left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell\right) + \left|m - n\right|}} \]

   4. Taylor expanded in K around 0 100.0%

      \[\leadsto \color{blue}{\cos \left(-M\right)} \cdot e^{\left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell\right) + \left|m - n\right|} \]
   5. Step-by-step derivation

      1. cos-neg 100.0%

         \[\leadsto \color{blue}{\cos M} \cdot e^{\left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell\right) + \left|m - n\right|} \]

   6. Simplified 100.0%

      \[\leadsto \color{blue}{\cos M} \cdot e^{\left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell\right) + \left|m - n\right|} \]

   7. Taylor expanded in m around inf 98.6%

      \[\leadsto \cos M \cdot e^{\color{blue}{-0.25 \cdot {m}^{2}}} \]
   8. Step-by-step derivation

      1. *-commutative 98.6%

         \[\leadsto \cos M \cdot e^{\color{blue}{{m}^{2} \cdot -0.25}} \]

      2. unpow2 98.6%

         \[\leadsto \cos M \cdot e^{\color{blue}{\left(m \cdot m\right)} \cdot -0.25} \]

      3. associate-*l* 98.6%

         \[\leadsto \cos M \cdot e^{\color{blue}{m \cdot \left(m \cdot -0.25\right)}} \]

   9. Simplified 98.6%

      \[\leadsto \cos M \cdot e^{\color{blue}{m \cdot \left(m \cdot -0.25\right)}} \]

   if -6e4 < m < 5.49999999999999995e-238

   1. Initial program 86.7%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 88.0%

         \[\leadsto \cos \left(\color{blue}{\frac{K}{\frac{2}{m + n}}} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]

      2. associate--r- 88.0%

         \[\leadsto \cos \left(\frac{K}{\frac{2}{m + n}} - M\right) \cdot e^{\color{blue}{\left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell\right) + \left|m - n\right|}} \]

   3. Simplified 88.0%

      \[\leadsto \color{blue}{\cos \left(\frac{K}{\frac{2}{m + n}} - M\right) \cdot e^{\left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell\right) + \left|m - n\right|}} \]

   4. Taylor expanded in K around 0 97.1%

      \[\leadsto \color{blue}{\cos \left(-M\right)} \cdot e^{\left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell\right) + \left|m - n\right|} \]
   5. Step-by-step derivation

      1. cos-neg 97.1%

         \[\leadsto \color{blue}{\cos M} \cdot e^{\left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell\right) + \left|m - n\right|} \]

   6. Simplified 97.1%

      \[\leadsto \color{blue}{\cos M} \cdot e^{\left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell\right) + \left|m - n\right|} \]

   7. Taylor expanded in M around inf 81.4%

      \[\leadsto \cos M \cdot e^{\left(\left(-\color{blue}{{M}^{2}}\right) - \ell\right) + \left|m - n\right|} \]
   8. Step-by-step derivation

      1. unpow2 81.4%

         \[\leadsto \cos M \cdot e^{\left(\left(-\color{blue}{M \cdot M}\right) - \ell\right) + \left|m - n\right|} \]

   9. Simplified 81.4%

      \[\leadsto \cos M \cdot e^{\left(\left(-\color{blue}{M \cdot M}\right) - \ell\right) + \left|m - n\right|} \]

   if 5.49999999999999995e-238 < m

   1. Initial program 66.8%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 65.9%

         \[\leadsto \cos \left(\color{blue}{\frac{K}{\frac{2}{m + n}}} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]

      2. associate--r- 65.9%

         \[\leadsto \cos \left(\frac{K}{\frac{2}{m + n}} - M\right) \cdot e^{\color{blue}{\left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell\right) + \left|m - n\right|}} \]

   3. Simplified 65.9%

      \[\leadsto \color{blue}{\cos \left(\frac{K}{\frac{2}{m + n}} - M\right) \cdot e^{\left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell\right) + \left|m - n\right|}} \]

   4. Taylor expanded in K around 0 96.5%

      \[\leadsto \color{blue}{\cos \left(-M\right)} \cdot e^{\left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell\right) + \left|m - n\right|} \]
   5. Step-by-step derivation

      1. cos-neg 96.5%

         \[\leadsto \color{blue}{\cos M} \cdot e^{\left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell\right) + \left|m - n\right|} \]

   6. Simplified 96.5%

      \[\leadsto \color{blue}{\cos M} \cdot e^{\left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell\right) + \left|m - n\right|} \]

   7. Taylor expanded in n around inf 56.7%

      \[\leadsto \cos M \cdot e^{\color{blue}{-0.25 \cdot {n}^{2}}} \]
   8. Step-by-step derivation

      1. *-commutative 56.7%

         \[\leadsto \cos M \cdot e^{\color{blue}{{n}^{2} \cdot -0.25}} \]

      2. unpow2 56.7%

         \[\leadsto \cos M \cdot e^{\color{blue}{\left(n \cdot n\right)} \cdot -0.25} \]

      3. associate-*l* 56.7%

         \[\leadsto \cos M \cdot e^{\color{blue}{n \cdot \left(n \cdot -0.25\right)}} \]

   9. Simplified 56.7%

      \[\leadsto \cos M \cdot e^{\color{blue}{n \cdot \left(n \cdot -0.25\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -60000:\\ \;\;\;\;\cos M \cdot e^{m \cdot \left(m \cdot -0.25\right)}\\ \mathbf{elif}\;m \leq 5.5 \cdot 10^{-238}:\\ \;\;\;\;\cos M \cdot e^{\left|m - n\right| - \left(\ell + M \cdot M\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{n \cdot \left(n \cdot -0.25\right)}\\ \end{array} \]

Alternative 3: 67.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\cos M}{e^{\ell}}\\ t_1 := {\left(e^{n}\right)}^{\left(n \cdot -0.25\right)}\\ \mathbf{if}\;n \leq -1.26 \cdot 10^{-24}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;n \leq -1.55 \cdot 10^{-213}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq 9.2 \cdot 10^{-232}:\\ \;\;\;\;\frac{\cos \left(\frac{m + n}{2} \cdot K - M\right)}{1 + m \cdot \left(m \cdot 0.25\right)}\\ \mathbf{elif}\;n \leq 510000:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (/ (cos M) (exp l))) (t_1 (pow (exp n) (* n -0.25))))
   (if (<= n -1.26e-24)
     t_1
     (if (<= n -1.55e-213)
       t_0
       (if (<= n 9.2e-232)
         (/ (cos (- (* (/ (+ m n) 2.0) K) M)) (+ 1.0 (* m (* m 0.25))))
         (if (<= n 510000.0) t_0 t_1))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = cos(M) / exp(l);
	double t_1 = pow(exp(n), (n * -0.25));
	double tmp;
	if (n <= -1.26e-24) {
		tmp = t_1;
	} else if (n <= -1.55e-213) {
		tmp = t_0;
	} else if (n <= 9.2e-232) {
		tmp = cos(((((m + n) / 2.0) * K) - M)) / (1.0 + (m * (m * 0.25)));
	} else if (n <= 510000.0) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos(m_1) / exp(l)
    t_1 = exp(n) ** (n * (-0.25d0))
    if (n <= (-1.26d-24)) then
        tmp = t_1
    else if (n <= (-1.55d-213)) then
        tmp = t_0
    else if (n <= 9.2d-232) then
        tmp = cos(((((m + n) / 2.0d0) * k) - m_1)) / (1.0d0 + (m * (m * 0.25d0)))
    else if (n <= 510000.0d0) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double t_0 = Math.cos(M) / Math.exp(l);
	double t_1 = Math.pow(Math.exp(n), (n * -0.25));
	double tmp;
	if (n <= -1.26e-24) {
		tmp = t_1;
	} else if (n <= -1.55e-213) {
		tmp = t_0;
	} else if (n <= 9.2e-232) {
		tmp = Math.cos(((((m + n) / 2.0) * K) - M)) / (1.0 + (m * (m * 0.25)));
	} else if (n <= 510000.0) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.cos(M) / math.exp(l)
	t_1 = math.pow(math.exp(n), (n * -0.25))
	tmp = 0
	if n <= -1.26e-24:
		tmp = t_1
	elif n <= -1.55e-213:
		tmp = t_0
	elif n <= 9.2e-232:
		tmp = math.cos(((((m + n) / 2.0) * K) - M)) / (1.0 + (m * (m * 0.25)))
	elif n <= 510000.0:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = Float64(cos(M) / exp(l))
	t_1 = exp(n) ^ Float64(n * -0.25)
	tmp = 0.0
	if (n <= -1.26e-24)
		tmp = t_1;
	elseif (n <= -1.55e-213)
		tmp = t_0;
	elseif (n <= 9.2e-232)
		tmp = Float64(cos(Float64(Float64(Float64(Float64(m + n) / 2.0) * K) - M)) / Float64(1.0 + Float64(m * Float64(m * 0.25))));
	elseif (n <= 510000.0)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = cos(M) / exp(l);
	t_1 = exp(n) ^ (n * -0.25);
	tmp = 0.0;
	if (n <= -1.26e-24)
		tmp = t_1;
	elseif (n <= -1.55e-213)
		tmp = t_0;
	elseif (n <= 9.2e-232)
		tmp = cos(((((m + n) / 2.0) * K) - M)) / (1.0 + (m * (m * 0.25)));
	elseif (n <= 510000.0)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(N[Cos[M], $MachinePrecision] / N[Exp[l], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Exp[n], $MachinePrecision], N[(n * -0.25), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[n, -1.26e-24], t$95$1, If[LessEqual[n, -1.55e-213], t$95$0, If[LessEqual[n, 9.2e-232], N[(N[Cos[N[(N[(N[(N[(m + n), $MachinePrecision] / 2.0), $MachinePrecision] * K), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision] / N[(1.0 + N[(m * N[(m * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 510000.0], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.25999999999999992e-24 or 5.1e5 < n

   1. Initial program 66.2%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 66.9%

         \[\leadsto \cos \left(\color{blue}{\frac{K}{\frac{2}{m + n}}} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]

      2. associate--r- 66.9%

         \[\leadsto \cos \left(\frac{K}{\frac{2}{m + n}} - M\right) \cdot e^{\color{blue}{\left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell\right) + \left|m - n\right|}} \]

   3. Simplified 66.9%

      \[\leadsto \color{blue}{\cos \left(\frac{K}{\frac{2}{m + n}} - M\right) \cdot e^{\left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell\right) + \left|m - n\right|}} \]

   4. Taylor expanded in K around 0 99.3%

      \[\leadsto \color{blue}{\cos \left(-M\right)} \cdot e^{\left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell\right) + \left|m - n\right|} \]
   5. Step-by-step derivation

      1. cos-neg 99.3%

         \[\leadsto \color{blue}{\cos M} \cdot e^{\left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell\right) + \left|m - n\right|} \]

   6. Simplified 99.3%

      \[\leadsto \color{blue}{\cos M} \cdot e^{\left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell\right) + \left|m - n\right|} \]

   7. Taylor expanded in n around inf 93.7%

      \[\leadsto \cos M \cdot e^{\color{blue}{-0.25 \cdot {n}^{2}}} \]
   8. Step-by-step derivation

      1. *-commutative 93.7%

         \[\leadsto \cos M \cdot e^{\color{blue}{{n}^{2} \cdot -0.25}} \]

      2. unpow2 93.7%

         \[\leadsto \cos M \cdot e^{\color{blue}{\left(n \cdot n\right)} \cdot -0.25} \]

      3. associate-*l* 93.7%

         \[\leadsto \cos M \cdot e^{\color{blue}{n \cdot \left(n \cdot -0.25\right)}} \]

   9. Simplified 93.7%

      \[\leadsto \cos M \cdot e^{\color{blue}{n \cdot \left(n \cdot -0.25\right)}} \]

   10. Taylor expanded in M around 0 93.7%

       \[\leadsto \color{blue}{e^{-0.25 \cdot {n}^{2}}} \]
   11. Step-by-step derivation

       1. *-commutative 93.7%

          \[\leadsto e^{\color{blue}{{n}^{2} \cdot -0.25}} \]

       2. unpow2 93.7%

          \[\leadsto e^{\color{blue}{\left(n \cdot n\right)} \cdot -0.25} \]

       3. associate-*r* 93.7%

          \[\leadsto e^{\color{blue}{n \cdot \left(n \cdot -0.25\right)}} \]

       4. exp-prod 93.7%

          \[\leadsto \color{blue}{{\left(e^{n}\right)}^{\left(n \cdot -0.25\right)}} \]

   12. Simplified 93.7%

       \[\leadsto \color{blue}{{\left(e^{n}\right)}^{\left(n \cdot -0.25\right)}} \]

   if -1.25999999999999992e-24 < n < -1.5499999999999999e-213 or 9.2e-232 < n < 5.1e5

   1. Initial program 85.0%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   2. Step-by-step derivation

      1. sub-neg 85.0%

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \color{blue}{\left(\ell + \left(-\left|m - n\right|\right)\right)}} \]

      2. associate--r+ 85.0%

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\color{blue}{\left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell\right) - \left(-\left|m - n\right|\right)}} \]

      3. exp-diff 52.6%

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \color{blue}{\frac{e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell}}{e^{-\left|m - n\right|}}} \]

      4. associate-*r/ 52.6%

         \[\leadsto \color{blue}{\frac{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell}}{e^{-\left|m - n\right|}}} \]

      5. associate-/l* 52.6%

         \[\leadsto \color{blue}{\frac{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right)}{\frac{e^{-\left|m - n\right|}}{e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell}}}} \]

      6. associate-*r/ 52.6%

         \[\leadsto \frac{\cos \left(\color{blue}{K \cdot \frac{m + n}{2}} - M\right)}{\frac{e^{-\left|m - n\right|}}{e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell}}} \]

      7. exp-diff 39.6%

         \[\leadsto \frac{\cos \left(K \cdot \frac{m + n}{2} - M\right)}{\frac{e^{-\left|m - n\right|}}{\color{blue}{\frac{e^{-{\left(\frac{m + n}{2} - M\right)}^{2}}}{e^{\ell}}}}} \]

   3. Simplified 85.0%

      \[\leadsto \color{blue}{\frac{\cos \left(K \cdot \frac{m + n}{2} - M\right)}{e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|n - m\right|\right)}}} \]

   4. Taylor expanded in l around inf 43.1%

      \[\leadsto \frac{\cos \left(K \cdot \frac{m + n}{2} - M\right)}{e^{\color{blue}{\ell}}} \]

   5. Taylor expanded in K around 0 43.0%

      \[\leadsto \frac{\color{blue}{\cos \left(-M\right)}}{e^{\ell}} \]
   6. Step-by-step derivation

      1. cos-neg 43.0%

         \[\leadsto \frac{\color{blue}{\cos M}}{e^{\ell}} \]

   7. Simplified 43.0%

      \[\leadsto \frac{\color{blue}{\cos M}}{e^{\ell}} \]

   if -1.5499999999999999e-213 < n < 9.2e-232

   1. Initial program 85.0%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   2. Step-by-step derivation

      1. sub-neg 85.0%

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \color{blue}{\left(\ell + \left(-\left|m - n\right|\right)\right)}} \]

      2. associate--r+ 85.0%

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\color{blue}{\left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell\right) - \left(-\left|m - n\right|\right)}} \]

      3. exp-diff 40.0%

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \color{blue}{\frac{e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell}}{e^{-\left|m - n\right|}}} \]

      4. associate-*r/ 40.0%

         \[\leadsto \color{blue}{\frac{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell}}{e^{-\left|m - n\right|}}} \]

      5. associate-/l* 40.0%

         \[\leadsto \color{blue}{\frac{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right)}{\frac{e^{-\left|m - n\right|}}{e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell}}}} \]

      6. associate-*r/ 40.0%

         \[\leadsto \frac{\cos \left(\color{blue}{K \cdot \frac{m + n}{2}} - M\right)}{\frac{e^{-\left|m - n\right|}}{e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell}}} \]

      7. exp-diff 32.5%

         \[\leadsto \frac{\cos \left(K \cdot \frac{m + n}{2} - M\right)}{\frac{e^{-\left|m - n\right|}}{\color{blue}{\frac{e^{-{\left(\frac{m + n}{2} - M\right)}^{2}}}{e^{\ell}}}}} \]

   3. Simplified 85.0%

      \[\leadsto \color{blue}{\frac{\cos \left(K \cdot \frac{m + n}{2} - M\right)}{e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|n - m\right|\right)}}} \]

   4. Taylor expanded in m around inf 60.7%

      \[\leadsto \frac{\cos \left(K \cdot \frac{m + n}{2} - M\right)}{e^{\color{blue}{0.25 \cdot {m}^{2}} + \left(\ell - \left|n - m\right|\right)}} \]
   5. Step-by-step derivation

      1. *-commutative 60.7%

         \[\leadsto \frac{\cos \left(K \cdot \frac{m + n}{2} - M\right)}{e^{\color{blue}{{m}^{2} \cdot 0.25} + \left(\ell - \left|n - m\right|\right)}} \]

      2. unpow2 60.7%

         \[\leadsto \frac{\cos \left(K \cdot \frac{m + n}{2} - M\right)}{e^{\color{blue}{\left(m \cdot m\right)} \cdot 0.25 + \left(\ell - \left|n - m\right|\right)}} \]

   6. Simplified 60.7%

      \[\leadsto \frac{\cos \left(K \cdot \frac{m + n}{2} - M\right)}{e^{\color{blue}{\left(m \cdot m\right) \cdot 0.25} + \left(\ell - \left|n - m\right|\right)}} \]

   7. Taylor expanded in m around inf 55.9%

      \[\leadsto \frac{\cos \left(K \cdot \frac{m + n}{2} - M\right)}{e^{\color{blue}{0.25 \cdot {m}^{2}}}} \]
   8. Step-by-step derivation

      1. *-commutative 55.9%

         \[\leadsto \frac{\cos \left(K \cdot \frac{m + n}{2} - M\right)}{e^{\color{blue}{{m}^{2} \cdot 0.25}}} \]

      2. unpow2 55.9%

         \[\leadsto \frac{\cos \left(K \cdot \frac{m + n}{2} - M\right)}{e^{\color{blue}{\left(m \cdot m\right)} \cdot 0.25}} \]

      3. associate-*r* 55.9%

         \[\leadsto \frac{\cos \left(K \cdot \frac{m + n}{2} - M\right)}{e^{\color{blue}{m \cdot \left(m \cdot 0.25\right)}}} \]

   9. Simplified 55.9%

      \[\leadsto \frac{\cos \left(K \cdot \frac{m + n}{2} - M\right)}{e^{\color{blue}{m \cdot \left(m \cdot 0.25\right)}}} \]

   10. Taylor expanded in m around 0 32.3%

       \[\leadsto \frac{\cos \left(K \cdot \frac{m + n}{2} - M\right)}{\color{blue}{1 + 0.25 \cdot {m}^{2}}} \]
   11. Step-by-step derivation

       1. *-commutative 32.3%

          \[\leadsto \frac{\cos \left(K \cdot \frac{m + n}{2} - M\right)}{1 + \color{blue}{{m}^{2} \cdot 0.25}} \]

       2. unpow2 32.3%

          \[\leadsto \frac{\cos \left(K \cdot \frac{m + n}{2} - M\right)}{1 + \color{blue}{\left(m \cdot m\right)} \cdot 0.25} \]

       3. associate-*r* 32.3%

          \[\leadsto \frac{\cos \left(K \cdot \frac{m + n}{2} - M\right)}{1 + \color{blue}{m \cdot \left(m \cdot 0.25\right)}} \]

   12. Simplified 32.3%

       \[\leadsto \frac{\cos \left(K \cdot \frac{m + n}{2} - M\right)}{\color{blue}{1 + m \cdot \left(m \cdot 0.25\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.26 \cdot 10^{-24}:\\ \;\;\;\;{\left(e^{n}\right)}^{\left(n \cdot -0.25\right)}\\ \mathbf{elif}\;n \leq -1.55 \cdot 10^{-213}:\\ \;\;\;\;\frac{\cos M}{e^{\ell}}\\ \mathbf{elif}\;n \leq 9.2 \cdot 10^{-232}:\\ \;\;\;\;\frac{\cos \left(\frac{m + n}{2} \cdot K - M\right)}{1 + m \cdot \left(m \cdot 0.25\right)}\\ \mathbf{elif}\;n \leq 510000:\\ \;\;\;\;\frac{\cos M}{e^{\ell}}\\ \mathbf{else}:\\ \;\;\;\;{\left(e^{n}\right)}^{\left(n \cdot -0.25\right)}\\ \end{array} \]

Alternative 4: 65.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -1550:\\ \;\;\;\;\cos M \cdot e^{m \cdot \left(m \cdot -0.25\right)}\\ \mathbf{elif}\;m \leq 9 \cdot 10^{-249}:\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(-M\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{n \cdot \left(n \cdot -0.25\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= m -1550.0)
   (* (cos M) (exp (* m (* m -0.25))))
   (if (<= m 9e-249)
     (* (cos M) (exp (* M (- M))))
     (* (cos M) (exp (* n (* n -0.25)))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -1550.0) {
		tmp = cos(M) * exp((m * (m * -0.25)));
	} else if (m <= 9e-249) {
		tmp = cos(M) * exp((M * -M));
	} else {
		tmp = cos(M) * exp((n * (n * -0.25)));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (m <= (-1550.0d0)) then
        tmp = cos(m_1) * exp((m * (m * (-0.25d0))))
    else if (m <= 9d-249) then
        tmp = cos(m_1) * exp((m_1 * -m_1))
    else
        tmp = cos(m_1) * exp((n * (n * (-0.25d0))))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -1550.0) {
		tmp = Math.cos(M) * Math.exp((m * (m * -0.25)));
	} else if (m <= 9e-249) {
		tmp = Math.cos(M) * Math.exp((M * -M));
	} else {
		tmp = Math.cos(M) * Math.exp((n * (n * -0.25)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if m <= -1550.0:
		tmp = math.cos(M) * math.exp((m * (m * -0.25)))
	elif m <= 9e-249:
		tmp = math.cos(M) * math.exp((M * -M))
	else:
		tmp = math.cos(M) * math.exp((n * (n * -0.25)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (m <= -1550.0)
		tmp = Float64(cos(M) * exp(Float64(m * Float64(m * -0.25))));
	elseif (m <= 9e-249)
		tmp = Float64(cos(M) * exp(Float64(M * Float64(-M))));
	else
		tmp = Float64(cos(M) * exp(Float64(n * Float64(n * -0.25))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (m <= -1550.0)
		tmp = cos(M) * exp((m * (m * -0.25)));
	elseif (m <= 9e-249)
		tmp = cos(M) * exp((M * -M));
	else
		tmp = cos(M) * exp((n * (n * -0.25)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[m, -1550.0], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(m * N[(m * -0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 9e-249], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(M * (-M)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(n * N[(n * -0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -1550

   1. Initial program 74.6%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 74.6%

         \[\leadsto \cos \left(\color{blue}{\frac{K}{\frac{2}{m + n}}} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]

      2. associate--r- 74.6%

         \[\leadsto \cos \left(\frac{K}{\frac{2}{m + n}} - M\right) \cdot e^{\color{blue}{\left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell\right) + \left|m - n\right|}} \]

   3. Simplified 74.6%

      \[\leadsto \color{blue}{\cos \left(\frac{K}{\frac{2}{m + n}} - M\right) \cdot e^{\left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell\right) + \left|m - n\right|}} \]

   4. Taylor expanded in K around 0 100.0%

      \[\leadsto \color{blue}{\cos \left(-M\right)} \cdot e^{\left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell\right) + \left|m - n\right|} \]
   5. Step-by-step derivation

      1. cos-neg 100.0%

         \[\leadsto \color{blue}{\cos M} \cdot e^{\left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell\right) + \left|m - n\right|} \]

   6. Simplified 100.0%

      \[\leadsto \color{blue}{\cos M} \cdot e^{\left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell\right) + \left|m - n\right|} \]

   7. Taylor expanded in m around inf 98.6%

      \[\leadsto \cos M \cdot e^{\color{blue}{-0.25 \cdot {m}^{2}}} \]
   8. Step-by-step derivation

      1. *-commutative 98.6%

         \[\leadsto \cos M \cdot e^{\color{blue}{{m}^{2} \cdot -0.25}} \]

      2. unpow2 98.6%

         \[\leadsto \cos M \cdot e^{\color{blue}{\left(m \cdot m\right)} \cdot -0.25} \]

      3. associate-*l* 98.6%

         \[\leadsto \cos M \cdot e^{\color{blue}{m \cdot \left(m \cdot -0.25\right)}} \]

   9. Simplified 98.6%

      \[\leadsto \cos M \cdot e^{\color{blue}{m \cdot \left(m \cdot -0.25\right)}} \]

   if -1550 < m < 8.99999999999999962e-249

   1. Initial program 86.1%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 87.5%

         \[\leadsto \cos \left(\color{blue}{\frac{K}{\frac{2}{m + n}}} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]

      2. associate--r- 87.5%

         \[\leadsto \cos \left(\frac{K}{\frac{2}{m + n}} - M\right) \cdot e^{\color{blue}{\left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell\right) + \left|m - n\right|}} \]

   3. Simplified 87.5%

      \[\leadsto \color{blue}{\cos \left(\frac{K}{\frac{2}{m + n}} - M\right) \cdot e^{\left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell\right) + \left|m - n\right|}} \]

   4. Taylor expanded in K around 0 97.0%

      \[\leadsto \color{blue}{\cos \left(-M\right)} \cdot e^{\left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell\right) + \left|m - n\right|} \]
   5. Step-by-step derivation

      1. cos-neg 97.0%

         \[\leadsto \color{blue}{\cos M} \cdot e^{\left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell\right) + \left|m - n\right|} \]

   6. Simplified 97.0%

      \[\leadsto \color{blue}{\cos M} \cdot e^{\left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell\right) + \left|m - n\right|} \]

   7. Taylor expanded in M around inf 72.8%

      \[\leadsto \cos M \cdot e^{\color{blue}{-1 \cdot {M}^{2}}} \]
   8. Step-by-step derivation

      1. mul-1-neg 72.8%

         \[\leadsto \cos M \cdot e^{\color{blue}{-{M}^{2}}} \]

      2. unpow2 72.8%

         \[\leadsto \cos M \cdot e^{-\color{blue}{M \cdot M}} \]

      3. distribute-rgt-neg-in 72.8%

         \[\leadsto \cos M \cdot e^{\color{blue}{M \cdot \left(-M\right)}} \]

   9. Simplified 72.8%

      \[\leadsto \cos M \cdot e^{\color{blue}{M \cdot \left(-M\right)}} \]

   if 8.99999999999999962e-249 < m

   1. Initial program 67.7%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 66.8%

         \[\leadsto \cos \left(\color{blue}{\frac{K}{\frac{2}{m + n}}} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]

      2. associate--r- 66.8%

         \[\leadsto \cos \left(\frac{K}{\frac{2}{m + n}} - M\right) \cdot e^{\color{blue}{\left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell\right) + \left|m - n\right|}} \]

   3. Simplified 66.8%

      \[\leadsto \color{blue}{\cos \left(\frac{K}{\frac{2}{m + n}} - M\right) \cdot e^{\left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell\right) + \left|m - n\right|}} \]

   4. Taylor expanded in K around 0 96.6%

      \[\leadsto \color{blue}{\cos \left(-M\right)} \cdot e^{\left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell\right) + \left|m - n\right|} \]
   5. Step-by-step derivation

      1. cos-neg 96.6%

         \[\leadsto \color{blue}{\cos M} \cdot e^{\left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell\right) + \left|m - n\right|} \]

   6. Simplified 96.6%

      \[\leadsto \color{blue}{\cos M} \cdot e^{\left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell\right) + \left|m - n\right|} \]

   7. Taylor expanded in n around inf 56.2%

      \[\leadsto \cos M \cdot e^{\color{blue}{-0.25 \cdot {n}^{2}}} \]
   8. Step-by-step derivation

      1. *-commutative 56.2%

         \[\leadsto \cos M \cdot e^{\color{blue}{{n}^{2} \cdot -0.25}} \]

      2. unpow2 56.2%

         \[\leadsto \cos M \cdot e^{\color{blue}{\left(n \cdot n\right)} \cdot -0.25} \]

      3. associate-*l* 56.2%

         \[\leadsto \cos M \cdot e^{\color{blue}{n \cdot \left(n \cdot -0.25\right)}} \]

   9. Simplified 56.2%

      \[\leadsto \cos M \cdot e^{\color{blue}{n \cdot \left(n \cdot -0.25\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 72.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -1550:\\ \;\;\;\;\cos M \cdot e^{m \cdot \left(m \cdot -0.25\right)}\\ \mathbf{elif}\;m \leq 9 \cdot 10^{-249}:\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(-M\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{n \cdot \left(n \cdot -0.25\right)}\\ \end{array} \]

Alternative 5: 76.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(e^{n}\right)}^{\left(n \cdot -0.25\right)}\\ \mathbf{if}\;n \leq -54:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq 2.8 \cdot 10^{-25}:\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(-M\right)}\\ \mathbf{elif}\;n \leq 510000:\\ \;\;\;\;\frac{\cos M}{e^{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (pow (exp n) (* n -0.25))))
   (if (<= n -54.0)
     t_0
     (if (<= n 2.8e-25)
       (* (cos M) (exp (* M (- M))))
       (if (<= n 510000.0) (/ (cos M) (exp l)) t_0)))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = pow(exp(n), (n * -0.25));
	double tmp;
	if (n <= -54.0) {
		tmp = t_0;
	} else if (n <= 2.8e-25) {
		tmp = cos(M) * exp((M * -M));
	} else if (n <= 510000.0) {
		tmp = cos(M) / exp(l);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(n) ** (n * (-0.25d0))
    if (n <= (-54.0d0)) then
        tmp = t_0
    else if (n <= 2.8d-25) then
        tmp = cos(m_1) * exp((m_1 * -m_1))
    else if (n <= 510000.0d0) then
        tmp = cos(m_1) / exp(l)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double t_0 = Math.pow(Math.exp(n), (n * -0.25));
	double tmp;
	if (n <= -54.0) {
		tmp = t_0;
	} else if (n <= 2.8e-25) {
		tmp = Math.cos(M) * Math.exp((M * -M));
	} else if (n <= 510000.0) {
		tmp = Math.cos(M) / Math.exp(l);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.pow(math.exp(n), (n * -0.25))
	tmp = 0
	if n <= -54.0:
		tmp = t_0
	elif n <= 2.8e-25:
		tmp = math.cos(M) * math.exp((M * -M))
	elif n <= 510000.0:
		tmp = math.cos(M) / math.exp(l)
	else:
		tmp = t_0
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = exp(n) ^ Float64(n * -0.25)
	tmp = 0.0
	if (n <= -54.0)
		tmp = t_0;
	elseif (n <= 2.8e-25)
		tmp = Float64(cos(M) * exp(Float64(M * Float64(-M))));
	elseif (n <= 510000.0)
		tmp = Float64(cos(M) / exp(l));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = exp(n) ^ (n * -0.25);
	tmp = 0.0;
	if (n <= -54.0)
		tmp = t_0;
	elseif (n <= 2.8e-25)
		tmp = cos(M) * exp((M * -M));
	elseif (n <= 510000.0)
		tmp = cos(M) / exp(l);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Power[N[Exp[n], $MachinePrecision], N[(n * -0.25), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[n, -54.0], t$95$0, If[LessEqual[n, 2.8e-25], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(M * (-M)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 510000.0], N[(N[Cos[M], $MachinePrecision] / N[Exp[l], $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if n < -54 or 5.1e5 < n

   1. Initial program 66.2%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 66.9%

         \[\leadsto \cos \left(\color{blue}{\frac{K}{\frac{2}{m + n}}} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]

      2. associate--r- 66.9%

         \[\leadsto \cos \left(\frac{K}{\frac{2}{m + n}} - M\right) \cdot e^{\color{blue}{\left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell\right) + \left|m - n\right|}} \]

   3. Simplified 66.9%

      \[\leadsto \color{blue}{\cos \left(\frac{K}{\frac{2}{m + n}} - M\right) \cdot e^{\left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell\right) + \left|m - n\right|}} \]

   4. Taylor expanded in K around 0 100.0%

      \[\leadsto \color{blue}{\cos \left(-M\right)} \cdot e^{\left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell\right) + \left|m - n\right|} \]
   5. Step-by-step derivation

      1. cos-neg 100.0%

         \[\leadsto \color{blue}{\cos M} \cdot e^{\left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell\right) + \left|m - n\right|} \]

   6. Simplified 100.0%

      \[\leadsto \color{blue}{\cos M} \cdot e^{\left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell\right) + \left|m - n\right|} \]

   7. Taylor expanded in n around inf 97.8%

      \[\leadsto \cos M \cdot e^{\color{blue}{-0.25 \cdot {n}^{2}}} \]
   8. Step-by-step derivation

      1. *-commutative 97.8%

         \[\leadsto \cos M \cdot e^{\color{blue}{{n}^{2} \cdot -0.25}} \]

      2. unpow2 97.8%

         \[\leadsto \cos M \cdot e^{\color{blue}{\left(n \cdot n\right)} \cdot -0.25} \]

      3. associate-*l* 97.8%

         \[\leadsto \cos M \cdot e^{\color{blue}{n \cdot \left(n \cdot -0.25\right)}} \]

   9. Simplified 97.8%

      \[\leadsto \cos M \cdot e^{\color{blue}{n \cdot \left(n \cdot -0.25\right)}} \]

   10. Taylor expanded in M around 0 97.8%

       \[\leadsto \color{blue}{e^{-0.25 \cdot {n}^{2}}} \]
   11. Step-by-step derivation

       1. *-commutative 97.8%

          \[\leadsto e^{\color{blue}{{n}^{2} \cdot -0.25}} \]

       2. unpow2 97.8%

          \[\leadsto e^{\color{blue}{\left(n \cdot n\right)} \cdot -0.25} \]

       3. associate-*r* 97.8%

          \[\leadsto e^{\color{blue}{n \cdot \left(n \cdot -0.25\right)}} \]

       4. exp-prod 97.8%

          \[\leadsto \color{blue}{{\left(e^{n}\right)}^{\left(n \cdot -0.25\right)}} \]

   12. Simplified 97.8%

       \[\leadsto \color{blue}{{\left(e^{n}\right)}^{\left(n \cdot -0.25\right)}} \]

   if -54 < n < 2.79999999999999988e-25

   1. Initial program 84.8%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 83.9%

         \[\leadsto \cos \left(\color{blue}{\frac{K}{\frac{2}{m + n}}} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]

      2. associate--r- 83.9%

         \[\leadsto \cos \left(\frac{K}{\frac{2}{m + n}} - M\right) \cdot e^{\color{blue}{\left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell\right) + \left|m - n\right|}} \]

   3. Simplified 83.9%

      \[\leadsto \color{blue}{\cos \left(\frac{K}{\frac{2}{m + n}} - M\right) \cdot e^{\left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell\right) + \left|m - n\right|}} \]

   4. Taylor expanded in K around 0 94.8%

      \[\leadsto \color{blue}{\cos \left(-M\right)} \cdot e^{\left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell\right) + \left|m - n\right|} \]
   5. Step-by-step derivation

      1. cos-neg 94.8%

         \[\leadsto \color{blue}{\cos M} \cdot e^{\left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell\right) + \left|m - n\right|} \]

   6. Simplified 94.8%

      \[\leadsto \color{blue}{\cos M} \cdot e^{\left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell\right) + \left|m - n\right|} \]

   7. Taylor expanded in M around inf 64.3%

      \[\leadsto \cos M \cdot e^{\color{blue}{-1 \cdot {M}^{2}}} \]
   8. Step-by-step derivation

      1. mul-1-neg 64.3%

         \[\leadsto \cos M \cdot e^{\color{blue}{-{M}^{2}}} \]

      2. unpow2 64.3%

         \[\leadsto \cos M \cdot e^{-\color{blue}{M \cdot M}} \]

      3. distribute-rgt-neg-in 64.3%

         \[\leadsto \cos M \cdot e^{\color{blue}{M \cdot \left(-M\right)}} \]

   9. Simplified 64.3%

      \[\leadsto \cos M \cdot e^{\color{blue}{M \cdot \left(-M\right)}} \]

   if 2.79999999999999988e-25 < n < 5.1e5

   1. Initial program 75.0%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   2. Step-by-step derivation

      1. sub-neg 75.0%

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \color{blue}{\left(\ell + \left(-\left|m - n\right|\right)\right)}} \]

      2. associate--r+ 75.0%

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\color{blue}{\left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell\right) - \left(-\left|m - n\right|\right)}} \]

      3. exp-diff 62.5%

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \color{blue}{\frac{e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell}}{e^{-\left|m - n\right|}}} \]

      4. associate-*r/ 62.5%

         \[\leadsto \color{blue}{\frac{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell}}{e^{-\left|m - n\right|}}} \]

      5. associate-/l* 62.5%

         \[\leadsto \color{blue}{\frac{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right)}{\frac{e^{-\left|m - n\right|}}{e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell}}}} \]

      6. associate-*r/ 62.5%

         \[\leadsto \frac{\cos \left(\color{blue}{K \cdot \frac{m + n}{2}} - M\right)}{\frac{e^{-\left|m - n\right|}}{e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell}}} \]

      7. exp-diff 25.0%

         \[\leadsto \frac{\cos \left(K \cdot \frac{m + n}{2} - M\right)}{\frac{e^{-\left|m - n\right|}}{\color{blue}{\frac{e^{-{\left(\frac{m + n}{2} - M\right)}^{2}}}{e^{\ell}}}}} \]

   3. Simplified 75.0%

      \[\leadsto \color{blue}{\frac{\cos \left(K \cdot \frac{m + n}{2} - M\right)}{e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|n - m\right|\right)}}} \]

   4. Taylor expanded in l around inf 75.0%

      \[\leadsto \frac{\cos \left(K \cdot \frac{m + n}{2} - M\right)}{e^{\color{blue}{\ell}}} \]

   5. Taylor expanded in K around 0 75.6%

      \[\leadsto \frac{\color{blue}{\cos \left(-M\right)}}{e^{\ell}} \]
   6. Step-by-step derivation

      1. cos-neg 75.6%

         \[\leadsto \frac{\color{blue}{\cos M}}{e^{\ell}} \]

   7. Simplified 75.6%

      \[\leadsto \frac{\color{blue}{\cos M}}{e^{\ell}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -54:\\ \;\;\;\;{\left(e^{n}\right)}^{\left(n \cdot -0.25\right)}\\ \mathbf{elif}\;n \leq 2.8 \cdot 10^{-25}:\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(-M\right)}\\ \mathbf{elif}\;n \leq 510000:\\ \;\;\;\;\frac{\cos M}{e^{\ell}}\\ \mathbf{else}:\\ \;\;\;\;{\left(e^{n}\right)}^{\left(n \cdot -0.25\right)}\\ \end{array} \]

Alternative 6: 65.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -1550:\\ \;\;\;\;\cos M \cdot e^{m \cdot \left(m \cdot -0.25\right)}\\ \mathbf{elif}\;m \leq 5 \cdot 10^{-237}:\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(-M\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(e^{n}\right)}^{\left(n \cdot -0.25\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= m -1550.0)
   (* (cos M) (exp (* m (* m -0.25))))
   (if (<= m 5e-237) (* (cos M) (exp (* M (- M)))) (pow (exp n) (* n -0.25)))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -1550.0) {
		tmp = cos(M) * exp((m * (m * -0.25)));
	} else if (m <= 5e-237) {
		tmp = cos(M) * exp((M * -M));
	} else {
		tmp = pow(exp(n), (n * -0.25));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (m <= (-1550.0d0)) then
        tmp = cos(m_1) * exp((m * (m * (-0.25d0))))
    else if (m <= 5d-237) then
        tmp = cos(m_1) * exp((m_1 * -m_1))
    else
        tmp = exp(n) ** (n * (-0.25d0))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -1550.0) {
		tmp = Math.cos(M) * Math.exp((m * (m * -0.25)));
	} else if (m <= 5e-237) {
		tmp = Math.cos(M) * Math.exp((M * -M));
	} else {
		tmp = Math.pow(Math.exp(n), (n * -0.25));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if m <= -1550.0:
		tmp = math.cos(M) * math.exp((m * (m * -0.25)))
	elif m <= 5e-237:
		tmp = math.cos(M) * math.exp((M * -M))
	else:
		tmp = math.pow(math.exp(n), (n * -0.25))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (m <= -1550.0)
		tmp = Float64(cos(M) * exp(Float64(m * Float64(m * -0.25))));
	elseif (m <= 5e-237)
		tmp = Float64(cos(M) * exp(Float64(M * Float64(-M))));
	else
		tmp = exp(n) ^ Float64(n * -0.25);
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (m <= -1550.0)
		tmp = cos(M) * exp((m * (m * -0.25)));
	elseif (m <= 5e-237)
		tmp = cos(M) * exp((M * -M));
	else
		tmp = exp(n) ^ (n * -0.25);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[m, -1550.0], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(m * N[(m * -0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 5e-237], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(M * (-M)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Power[N[Exp[n], $MachinePrecision], N[(n * -0.25), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -1550

   1. Initial program 74.6%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 74.6%

         \[\leadsto \cos \left(\color{blue}{\frac{K}{\frac{2}{m + n}}} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]

      2. associate--r- 74.6%

         \[\leadsto \cos \left(\frac{K}{\frac{2}{m + n}} - M\right) \cdot e^{\color{blue}{\left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell\right) + \left|m - n\right|}} \]

   3. Simplified 74.6%

      \[\leadsto \color{blue}{\cos \left(\frac{K}{\frac{2}{m + n}} - M\right) \cdot e^{\left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell\right) + \left|m - n\right|}} \]

   4. Taylor expanded in K around 0 100.0%

      \[\leadsto \color{blue}{\cos \left(-M\right)} \cdot e^{\left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell\right) + \left|m - n\right|} \]
   5. Step-by-step derivation

      1. cos-neg 100.0%

         \[\leadsto \color{blue}{\cos M} \cdot e^{\left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell\right) + \left|m - n\right|} \]

   6. Simplified 100.0%

      \[\leadsto \color{blue}{\cos M} \cdot e^{\left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell\right) + \left|m - n\right|} \]

   7. Taylor expanded in m around inf 98.6%

      \[\leadsto \cos M \cdot e^{\color{blue}{-0.25 \cdot {m}^{2}}} \]
   8. Step-by-step derivation

      1. *-commutative 98.6%

         \[\leadsto \cos M \cdot e^{\color{blue}{{m}^{2} \cdot -0.25}} \]

      2. unpow2 98.6%

         \[\leadsto \cos M \cdot e^{\color{blue}{\left(m \cdot m\right)} \cdot -0.25} \]

      3. associate-*l* 98.6%

         \[\leadsto \cos M \cdot e^{\color{blue}{m \cdot \left(m \cdot -0.25\right)}} \]

   9. Simplified 98.6%

      \[\leadsto \cos M \cdot e^{\color{blue}{m \cdot \left(m \cdot -0.25\right)}} \]

   if -1550 < m < 5.0000000000000002e-237

   1. Initial program 86.7%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 88.0%

         \[\leadsto \cos \left(\color{blue}{\frac{K}{\frac{2}{m + n}}} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]

      2. associate--r- 88.0%

         \[\leadsto \cos \left(\frac{K}{\frac{2}{m + n}} - M\right) \cdot e^{\color{blue}{\left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell\right) + \left|m - n\right|}} \]

   3. Simplified 88.0%

      \[\leadsto \color{blue}{\cos \left(\frac{K}{\frac{2}{m + n}} - M\right) \cdot e^{\left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell\right) + \left|m - n\right|}} \]

   4. Taylor expanded in K around 0 97.1%

      \[\leadsto \color{blue}{\cos \left(-M\right)} \cdot e^{\left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell\right) + \left|m - n\right|} \]
   5. Step-by-step derivation

      1. cos-neg 97.1%

         \[\leadsto \color{blue}{\cos M} \cdot e^{\left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell\right) + \left|m - n\right|} \]

   6. Simplified 97.1%

      \[\leadsto \color{blue}{\cos M} \cdot e^{\left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell\right) + \left|m - n\right|} \]

   7. Taylor expanded in M around inf 71.3%

      \[\leadsto \cos M \cdot e^{\color{blue}{-1 \cdot {M}^{2}}} \]
   8. Step-by-step derivation

      1. mul-1-neg 71.3%

         \[\leadsto \cos M \cdot e^{\color{blue}{-{M}^{2}}} \]

      2. unpow2 71.3%

         \[\leadsto \cos M \cdot e^{-\color{blue}{M \cdot M}} \]

      3. distribute-rgt-neg-in 71.3%

         \[\leadsto \cos M \cdot e^{\color{blue}{M \cdot \left(-M\right)}} \]

   9. Simplified 71.3%

      \[\leadsto \cos M \cdot e^{\color{blue}{M \cdot \left(-M\right)}} \]

   if 5.0000000000000002e-237 < m

   1. Initial program 66.8%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 65.9%

         \[\leadsto \cos \left(\color{blue}{\frac{K}{\frac{2}{m + n}}} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]

      2. associate--r- 65.9%

         \[\leadsto \cos \left(\frac{K}{\frac{2}{m + n}} - M\right) \cdot e^{\color{blue}{\left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell\right) + \left|m - n\right|}} \]

   3. Simplified 65.9%

      \[\leadsto \color{blue}{\cos \left(\frac{K}{\frac{2}{m + n}} - M\right) \cdot e^{\left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell\right) + \left|m - n\right|}} \]

   4. Taylor expanded in K around 0 96.5%

      \[\leadsto \color{blue}{\cos \left(-M\right)} \cdot e^{\left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell\right) + \left|m - n\right|} \]
   5. Step-by-step derivation

      1. cos-neg 96.5%

         \[\leadsto \color{blue}{\cos M} \cdot e^{\left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell\right) + \left|m - n\right|} \]

   6. Simplified 96.5%

      \[\leadsto \color{blue}{\cos M} \cdot e^{\left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell\right) + \left|m - n\right|} \]

   7. Taylor expanded in n around inf 56.7%

      \[\leadsto \cos M \cdot e^{\color{blue}{-0.25 \cdot {n}^{2}}} \]
   8. Step-by-step derivation

      1. *-commutative 56.7%

         \[\leadsto \cos M \cdot e^{\color{blue}{{n}^{2} \cdot -0.25}} \]

      2. unpow2 56.7%

         \[\leadsto \cos M \cdot e^{\color{blue}{\left(n \cdot n\right)} \cdot -0.25} \]

      3. associate-*l* 56.7%

         \[\leadsto \cos M \cdot e^{\color{blue}{n \cdot \left(n \cdot -0.25\right)}} \]

   9. Simplified 56.7%

      \[\leadsto \cos M \cdot e^{\color{blue}{n \cdot \left(n \cdot -0.25\right)}} \]

   10. Taylor expanded in M around 0 56.7%

       \[\leadsto \color{blue}{e^{-0.25 \cdot {n}^{2}}} \]
   11. Step-by-step derivation

       1. *-commutative 56.7%

          \[\leadsto e^{\color{blue}{{n}^{2} \cdot -0.25}} \]

       2. unpow2 56.7%

          \[\leadsto e^{\color{blue}{\left(n \cdot n\right)} \cdot -0.25} \]

       3. associate-*r* 56.7%

          \[\leadsto e^{\color{blue}{n \cdot \left(n \cdot -0.25\right)}} \]

       4. exp-prod 56.7%

          \[\leadsto \color{blue}{{\left(e^{n}\right)}^{\left(n \cdot -0.25\right)}} \]

   12. Simplified 56.7%

       \[\leadsto \color{blue}{{\left(e^{n}\right)}^{\left(n \cdot -0.25\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 72.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -1550:\\ \;\;\;\;\cos M \cdot e^{m \cdot \left(m \cdot -0.25\right)}\\ \mathbf{elif}\;m \leq 5 \cdot 10^{-237}:\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(-M\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(e^{n}\right)}^{\left(n \cdot -0.25\right)}\\ \end{array} \]

Alternative 7: 39.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -8.5 \cdot 10^{+164}:\\ \;\;\;\;\frac{\cos \left(\frac{m + n}{2} \cdot K - M\right)}{1 + m \cdot \left(m \cdot 0.25\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos M}{e^{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= m -8.5e+164)
   (/ (cos (- (* (/ (+ m n) 2.0) K) M)) (+ 1.0 (* m (* m 0.25))))
   (/ (cos M) (exp l))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -8.5e+164) {
		tmp = cos(((((m + n) / 2.0) * K) - M)) / (1.0 + (m * (m * 0.25)));
	} else {
		tmp = cos(M) / exp(l);
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (m <= (-8.5d+164)) then
        tmp = cos(((((m + n) / 2.0d0) * k) - m_1)) / (1.0d0 + (m * (m * 0.25d0)))
    else
        tmp = cos(m_1) / exp(l)
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -8.5e+164) {
		tmp = Math.cos(((((m + n) / 2.0) * K) - M)) / (1.0 + (m * (m * 0.25)));
	} else {
		tmp = Math.cos(M) / Math.exp(l);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if m <= -8.5e+164:
		tmp = math.cos(((((m + n) / 2.0) * K) - M)) / (1.0 + (m * (m * 0.25)))
	else:
		tmp = math.cos(M) / math.exp(l)
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (m <= -8.5e+164)
		tmp = Float64(cos(Float64(Float64(Float64(Float64(m + n) / 2.0) * K) - M)) / Float64(1.0 + Float64(m * Float64(m * 0.25))));
	else
		tmp = Float64(cos(M) / exp(l));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (m <= -8.5e+164)
		tmp = cos(((((m + n) / 2.0) * K) - M)) / (1.0 + (m * (m * 0.25)));
	else
		tmp = cos(M) / exp(l);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[m, -8.5e+164], N[(N[Cos[N[(N[(N[(N[(m + n), $MachinePrecision] / 2.0), $MachinePrecision] * K), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision] / N[(1.0 + N[(m * N[(m * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] / N[Exp[l], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -8.50000000000000027e164

   1. Initial program 73.2%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   2. Step-by-step derivation

      1. sub-neg 73.2%

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \color{blue}{\left(\ell + \left(-\left|m - n\right|\right)\right)}} \]

      2. associate--r+ 73.2%

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\color{blue}{\left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell\right) - \left(-\left|m - n\right|\right)}} \]

      3. exp-diff 0.0%

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \color{blue}{\frac{e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell}}{e^{-\left|m - n\right|}}} \]

      4. associate-*r/ 0.0%

         \[\leadsto \color{blue}{\frac{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell}}{e^{-\left|m - n\right|}}} \]

      5. associate-/l* 0.0%

         \[\leadsto \color{blue}{\frac{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right)}{\frac{e^{-\left|m - n\right|}}{e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell}}}} \]

      6. associate-*r/ 0.0%

         \[\leadsto \frac{\cos \left(\color{blue}{K \cdot \frac{m + n}{2}} - M\right)}{\frac{e^{-\left|m - n\right|}}{e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell}}} \]

      7. exp-diff 0.0%

         \[\leadsto \frac{\cos \left(K \cdot \frac{m + n}{2} - M\right)}{\frac{e^{-\left|m - n\right|}}{\color{blue}{\frac{e^{-{\left(\frac{m + n}{2} - M\right)}^{2}}}{e^{\ell}}}}} \]

   3. Simplified 73.2%

      \[\leadsto \color{blue}{\frac{\cos \left(K \cdot \frac{m + n}{2} - M\right)}{e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|n - m\right|\right)}}} \]

   4. Taylor expanded in m around inf 73.2%

      \[\leadsto \frac{\cos \left(K \cdot \frac{m + n}{2} - M\right)}{e^{\color{blue}{0.25 \cdot {m}^{2}} + \left(\ell - \left|n - m\right|\right)}} \]
   5. Step-by-step derivation

      1. *-commutative 73.2%

         \[\leadsto \frac{\cos \left(K \cdot \frac{m + n}{2} - M\right)}{e^{\color{blue}{{m}^{2} \cdot 0.25} + \left(\ell - \left|n - m\right|\right)}} \]

      2. unpow2 73.2%

         \[\leadsto \frac{\cos \left(K \cdot \frac{m + n}{2} - M\right)}{e^{\color{blue}{\left(m \cdot m\right)} \cdot 0.25 + \left(\ell - \left|n - m\right|\right)}} \]

   6. Simplified 73.2%

      \[\leadsto \frac{\cos \left(K \cdot \frac{m + n}{2} - M\right)}{e^{\color{blue}{\left(m \cdot m\right) \cdot 0.25} + \left(\ell - \left|n - m\right|\right)}} \]

   7. Taylor expanded in m around inf 73.2%

      \[\leadsto \frac{\cos \left(K \cdot \frac{m + n}{2} - M\right)}{e^{\color{blue}{0.25 \cdot {m}^{2}}}} \]
   8. Step-by-step derivation

      1. *-commutative 73.2%

         \[\leadsto \frac{\cos \left(K \cdot \frac{m + n}{2} - M\right)}{e^{\color{blue}{{m}^{2} \cdot 0.25}}} \]

      2. unpow2 73.2%

         \[\leadsto \frac{\cos \left(K \cdot \frac{m + n}{2} - M\right)}{e^{\color{blue}{\left(m \cdot m\right)} \cdot 0.25}} \]

      3. associate-*r* 73.2%

         \[\leadsto \frac{\cos \left(K \cdot \frac{m + n}{2} - M\right)}{e^{\color{blue}{m \cdot \left(m \cdot 0.25\right)}}} \]

   9. Simplified 73.2%

      \[\leadsto \frac{\cos \left(K \cdot \frac{m + n}{2} - M\right)}{e^{\color{blue}{m \cdot \left(m \cdot 0.25\right)}}} \]

   10. Taylor expanded in m around 0 73.2%

       \[\leadsto \frac{\cos \left(K \cdot \frac{m + n}{2} - M\right)}{\color{blue}{1 + 0.25 \cdot {m}^{2}}} \]
   11. Step-by-step derivation

       1. *-commutative 73.2%

          \[\leadsto \frac{\cos \left(K \cdot \frac{m + n}{2} - M\right)}{1 + \color{blue}{{m}^{2} \cdot 0.25}} \]

       2. unpow2 73.2%

          \[\leadsto \frac{\cos \left(K \cdot \frac{m + n}{2} - M\right)}{1 + \color{blue}{\left(m \cdot m\right)} \cdot 0.25} \]

       3. associate-*r* 73.2%

          \[\leadsto \frac{\cos \left(K \cdot \frac{m + n}{2} - M\right)}{1 + \color{blue}{m \cdot \left(m \cdot 0.25\right)}} \]

   12. Simplified 73.2%

       \[\leadsto \frac{\cos \left(K \cdot \frac{m + n}{2} - M\right)}{\color{blue}{1 + m \cdot \left(m \cdot 0.25\right)}} \]

   if -8.50000000000000027e164 < m

   1. Initial program 75.1%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   2. Step-by-step derivation

      1. sub-neg 75.1%

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \color{blue}{\left(\ell + \left(-\left|m - n\right|\right)\right)}} \]

      2. associate--r+ 75.1%

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\color{blue}{\left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell\right) - \left(-\left|m - n\right|\right)}} \]

      3. exp-diff 28.1%

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \color{blue}{\frac{e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell}}{e^{-\left|m - n\right|}}} \]

      4. associate-*r/ 28.1%

         \[\leadsto \color{blue}{\frac{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell}}{e^{-\left|m - n\right|}}} \]

      5. associate-/l* 28.1%

         \[\leadsto \color{blue}{\frac{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right)}{\frac{e^{-\left|m - n\right|}}{e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell}}}} \]

      6. associate-*r/ 28.1%

         \[\leadsto \frac{\cos \left(\color{blue}{K \cdot \frac{m + n}{2}} - M\right)}{\frac{e^{-\left|m - n\right|}}{e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell}}} \]

      7. exp-diff 20.7%

         \[\leadsto \frac{\cos \left(K \cdot \frac{m + n}{2} - M\right)}{\frac{e^{-\left|m - n\right|}}{\color{blue}{\frac{e^{-{\left(\frac{m + n}{2} - M\right)}^{2}}}{e^{\ell}}}}} \]

   3. Simplified 75.1%

      \[\leadsto \color{blue}{\frac{\cos \left(K \cdot \frac{m + n}{2} - M\right)}{e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|n - m\right|\right)}}} \]

   4. Taylor expanded in l around inf 32.4%

      \[\leadsto \frac{\cos \left(K \cdot \frac{m + n}{2} - M\right)}{e^{\color{blue}{\ell}}} \]

   5. Taylor expanded in K around 0 36.9%

      \[\leadsto \frac{\color{blue}{\cos \left(-M\right)}}{e^{\ell}} \]
   6. Step-by-step derivation

      1. cos-neg 36.9%

         \[\leadsto \frac{\color{blue}{\cos M}}{e^{\ell}} \]

   7. Simplified 36.9%

      \[\leadsto \frac{\color{blue}{\cos M}}{e^{\ell}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 42.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -8.5 \cdot 10^{+164}:\\ \;\;\;\;\frac{\cos \left(\frac{m + n}{2} \cdot K - M\right)}{1 + m \cdot \left(m \cdot 0.25\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos M}{e^{\ell}}\\ \end{array} \]

Alternative 8: 39.2% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -8.5 \cdot 10^{+164}:\\ \;\;\;\;\frac{\cos \left(\frac{m + n}{2} \cdot K - M\right)}{1 + m \cdot \left(m \cdot 0.25\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{-\ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= m -8.5e+164)
   (/ (cos (- (* (/ (+ m n) 2.0) K) M)) (+ 1.0 (* m (* m 0.25))))
   (exp (- l))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -8.5e+164) {
		tmp = cos(((((m + n) / 2.0) * K) - M)) / (1.0 + (m * (m * 0.25)));
	} else {
		tmp = exp(-l);
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (m <= (-8.5d+164)) then
        tmp = cos(((((m + n) / 2.0d0) * k) - m_1)) / (1.0d0 + (m * (m * 0.25d0)))
    else
        tmp = exp(-l)
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -8.5e+164) {
		tmp = Math.cos(((((m + n) / 2.0) * K) - M)) / (1.0 + (m * (m * 0.25)));
	} else {
		tmp = Math.exp(-l);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if m <= -8.5e+164:
		tmp = math.cos(((((m + n) / 2.0) * K) - M)) / (1.0 + (m * (m * 0.25)))
	else:
		tmp = math.exp(-l)
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (m <= -8.5e+164)
		tmp = Float64(cos(Float64(Float64(Float64(Float64(m + n) / 2.0) * K) - M)) / Float64(1.0 + Float64(m * Float64(m * 0.25))));
	else
		tmp = exp(Float64(-l));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (m <= -8.5e+164)
		tmp = cos(((((m + n) / 2.0) * K) - M)) / (1.0 + (m * (m * 0.25)));
	else
		tmp = exp(-l);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[m, -8.5e+164], N[(N[Cos[N[(N[(N[(N[(m + n), $MachinePrecision] / 2.0), $MachinePrecision] * K), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision] / N[(1.0 + N[(m * N[(m * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Exp[(-l)], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -8.50000000000000027e164

   1. Initial program 73.2%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   2. Step-by-step derivation

      1. sub-neg 73.2%

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \color{blue}{\left(\ell + \left(-\left|m - n\right|\right)\right)}} \]

      2. associate--r+ 73.2%

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\color{blue}{\left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell\right) - \left(-\left|m - n\right|\right)}} \]

      3. exp-diff 0.0%

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \color{blue}{\frac{e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell}}{e^{-\left|m - n\right|}}} \]

      4. associate-*r/ 0.0%

         \[\leadsto \color{blue}{\frac{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell}}{e^{-\left|m - n\right|}}} \]

      5. associate-/l* 0.0%

         \[\leadsto \color{blue}{\frac{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right)}{\frac{e^{-\left|m - n\right|}}{e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell}}}} \]

      6. associate-*r/ 0.0%

         \[\leadsto \frac{\cos \left(\color{blue}{K \cdot \frac{m + n}{2}} - M\right)}{\frac{e^{-\left|m - n\right|}}{e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell}}} \]

      7. exp-diff 0.0%

         \[\leadsto \frac{\cos \left(K \cdot \frac{m + n}{2} - M\right)}{\frac{e^{-\left|m - n\right|}}{\color{blue}{\frac{e^{-{\left(\frac{m + n}{2} - M\right)}^{2}}}{e^{\ell}}}}} \]

   3. Simplified 73.2%

      \[\leadsto \color{blue}{\frac{\cos \left(K \cdot \frac{m + n}{2} - M\right)}{e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|n - m\right|\right)}}} \]

   4. Taylor expanded in m around inf 73.2%

      \[\leadsto \frac{\cos \left(K \cdot \frac{m + n}{2} - M\right)}{e^{\color{blue}{0.25 \cdot {m}^{2}} + \left(\ell - \left|n - m\right|\right)}} \]
   5. Step-by-step derivation

      1. *-commutative 73.2%

         \[\leadsto \frac{\cos \left(K \cdot \frac{m + n}{2} - M\right)}{e^{\color{blue}{{m}^{2} \cdot 0.25} + \left(\ell - \left|n - m\right|\right)}} \]

      2. unpow2 73.2%

         \[\leadsto \frac{\cos \left(K \cdot \frac{m + n}{2} - M\right)}{e^{\color{blue}{\left(m \cdot m\right)} \cdot 0.25 + \left(\ell - \left|n - m\right|\right)}} \]

   6. Simplified 73.2%

      \[\leadsto \frac{\cos \left(K \cdot \frac{m + n}{2} - M\right)}{e^{\color{blue}{\left(m \cdot m\right) \cdot 0.25} + \left(\ell - \left|n - m\right|\right)}} \]

   7. Taylor expanded in m around inf 73.2%

      \[\leadsto \frac{\cos \left(K \cdot \frac{m + n}{2} - M\right)}{e^{\color{blue}{0.25 \cdot {m}^{2}}}} \]
   8. Step-by-step derivation

      1. *-commutative 73.2%

         \[\leadsto \frac{\cos \left(K \cdot \frac{m + n}{2} - M\right)}{e^{\color{blue}{{m}^{2} \cdot 0.25}}} \]

      2. unpow2 73.2%

         \[\leadsto \frac{\cos \left(K \cdot \frac{m + n}{2} - M\right)}{e^{\color{blue}{\left(m \cdot m\right)} \cdot 0.25}} \]

      3. associate-*r* 73.2%

         \[\leadsto \frac{\cos \left(K \cdot \frac{m + n}{2} - M\right)}{e^{\color{blue}{m \cdot \left(m \cdot 0.25\right)}}} \]

   9. Simplified 73.2%

      \[\leadsto \frac{\cos \left(K \cdot \frac{m + n}{2} - M\right)}{e^{\color{blue}{m \cdot \left(m \cdot 0.25\right)}}} \]

   10. Taylor expanded in m around 0 73.2%

       \[\leadsto \frac{\cos \left(K \cdot \frac{m + n}{2} - M\right)}{\color{blue}{1 + 0.25 \cdot {m}^{2}}} \]
   11. Step-by-step derivation

       1. *-commutative 73.2%

          \[\leadsto \frac{\cos \left(K \cdot \frac{m + n}{2} - M\right)}{1 + \color{blue}{{m}^{2} \cdot 0.25}} \]

       2. unpow2 73.2%

          \[\leadsto \frac{\cos \left(K \cdot \frac{m + n}{2} - M\right)}{1 + \color{blue}{\left(m \cdot m\right)} \cdot 0.25} \]

       3. associate-*r* 73.2%

          \[\leadsto \frac{\cos \left(K \cdot \frac{m + n}{2} - M\right)}{1 + \color{blue}{m \cdot \left(m \cdot 0.25\right)}} \]

   12. Simplified 73.2%

       \[\leadsto \frac{\cos \left(K \cdot \frac{m + n}{2} - M\right)}{\color{blue}{1 + m \cdot \left(m \cdot 0.25\right)}} \]

   if -8.50000000000000027e164 < m

   1. Initial program 75.1%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
   2. Step-by-step derivation

      1. sub-neg 75.1%

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \color{blue}{\left(\ell + \left(-\left|m - n\right|\right)\right)}} \]

      2. associate--r+ 75.1%

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\color{blue}{\left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell\right) - \left(-\left|m - n\right|\right)}} \]

      3. exp-diff 28.1%

         \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \color{blue}{\frac{e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell}}{e^{-\left|m - n\right|}}} \]

      4. associate-*r/ 28.1%

         \[\leadsto \color{blue}{\frac{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell}}{e^{-\left|m - n\right|}}} \]

      5. associate-/l* 28.1%

         \[\leadsto \color{blue}{\frac{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right)}{\frac{e^{-\left|m - n\right|}}{e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell}}}} \]

      6. associate-*r/ 28.1%

         \[\leadsto \frac{\cos \left(\color{blue}{K \cdot \frac{m + n}{2}} - M\right)}{\frac{e^{-\left|m - n\right|}}{e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell}}} \]

      7. exp-diff 20.7%

         \[\leadsto \frac{\cos \left(K \cdot \frac{m + n}{2} - M\right)}{\frac{e^{-\left|m - n\right|}}{\color{blue}{\frac{e^{-{\left(\frac{m + n}{2} - M\right)}^{2}}}{e^{\ell}}}}} \]

   3. Simplified 75.1%

      \[\leadsto \color{blue}{\frac{\cos \left(K \cdot \frac{m + n}{2} - M\right)}{e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|n - m\right|\right)}}} \]

   4. Taylor expanded in l around inf 32.4%

      \[\leadsto \frac{\cos \left(K \cdot \frac{m + n}{2} - M\right)}{e^{\color{blue}{\ell}}} \]

   5. Taylor expanded in K around 0 36.9%

      \[\leadsto \frac{\color{blue}{\cos \left(-M\right)}}{e^{\ell}} \]
   6. Step-by-step derivation

      1. cos-neg 36.9%

         \[\leadsto \frac{\color{blue}{\cos M}}{e^{\ell}} \]

   7. Simplified 36.9%

      \[\leadsto \frac{\color{blue}{\cos M}}{e^{\ell}} \]

   8. Taylor expanded in M around 0 36.5%

      \[\leadsto \color{blue}{\frac{1}{e^{\ell}}} \]
   9. Step-by-step derivation

      1. rec-exp 36.5%

         \[\leadsto \color{blue}{e^{-\ell}} \]

   10. Simplified 36.5%

       \[\leadsto \color{blue}{e^{-\ell}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 42.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -8.5 \cdot 10^{+164}:\\ \;\;\;\;\frac{\cos \left(\frac{m + n}{2} \cdot K - M\right)}{1 + m \cdot \left(m \cdot 0.25\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{-\ell}\\ \end{array} \]

Alternative 9: 35.1% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ e^{-\ell} \end{array} \]

FPCore

(FPCore (K m n M l) :precision binary64 (exp (- l)))

C

double code(double K, double m, double n, double M, double l) {
	return exp(-l);
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    code = exp(-l)
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	return Math.exp(-l);
}

Python

def code(K, m, n, M, l):
	return math.exp(-l)

Julia

function code(K, m, n, M, l)
	return exp(Float64(-l))
end

MATLAB

function tmp = code(K, m, n, M, l)
	tmp = exp(-l);
end

Wolfram

code[K_, m_, n_, M_, l_] := N[Exp[(-l)], $MachinePrecision]

Derivation

1. Initial program 74.8%

   \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
2. Step-by-step derivation

   1. sub-neg 74.8%

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \color{blue}{\left(\ell + \left(-\left|m - n\right|\right)\right)}} \]

   2. associate--r+ 74.8%

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\color{blue}{\left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell\right) - \left(-\left|m - n\right|\right)}} \]

   3. exp-diff 23.6%

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \color{blue}{\frac{e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell}}{e^{-\left|m - n\right|}}} \]

   4. associate-*r/ 23.6%

      \[\leadsto \color{blue}{\frac{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell}}{e^{-\left|m - n\right|}}} \]

   5. associate-/l* 23.6%

      \[\leadsto \color{blue}{\frac{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right)}{\frac{e^{-\left|m - n\right|}}{e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell}}}} \]

   6. associate-*r/ 23.6%

      \[\leadsto \frac{\cos \left(\color{blue}{K \cdot \frac{m + n}{2}} - M\right)}{\frac{e^{-\left|m - n\right|}}{e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell}}} \]

   7. exp-diff 17.4%

      \[\leadsto \frac{\cos \left(K \cdot \frac{m + n}{2} - M\right)}{\frac{e^{-\left|m - n\right|}}{\color{blue}{\frac{e^{-{\left(\frac{m + n}{2} - M\right)}^{2}}}{e^{\ell}}}}} \]

3. Simplified 74.8%

   \[\leadsto \color{blue}{\frac{\cos \left(K \cdot \frac{m + n}{2} - M\right)}{e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|n - m\right|\right)}}} \]

4. Taylor expanded in l around inf 31.0%

   \[\leadsto \frac{\cos \left(K \cdot \frac{m + n}{2} - M\right)}{e^{\color{blue}{\ell}}} \]

5. Taylor expanded in K around 0 36.0%

   \[\leadsto \frac{\color{blue}{\cos \left(-M\right)}}{e^{\ell}} \]
6. Step-by-step derivation

   1. cos-neg 36.0%

      \[\leadsto \frac{\color{blue}{\cos M}}{e^{\ell}} \]

7. Simplified 36.0%

   \[\leadsto \frac{\color{blue}{\cos M}}{e^{\ell}} \]

8. Taylor expanded in M around 0 35.6%

   \[\leadsto \color{blue}{\frac{1}{e^{\ell}}} \]
9. Step-by-step derivation

   1. rec-exp 35.6%

      \[\leadsto \color{blue}{e^{-\ell}} \]

10. Simplified 35.6%

    \[\leadsto \color{blue}{e^{-\ell}} \]

11. Final simplification 35.6%

    \[\leadsto e^{-\ell} \]

Alternative 10: 6.8% accurate, 141.7× speedup

Math

\[\begin{array}{l} \\ 1 - \ell \end{array} \]

FPCore

(FPCore (K m n M l) :precision binary64 (- 1.0 l))

C

double code(double K, double m, double n, double M, double l) {
	return 1.0 - l;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    code = 1.0d0 - l
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	return 1.0 - l;
}

Python

def code(K, m, n, M, l):
	return 1.0 - l

Julia

function code(K, m, n, M, l)
	return Float64(1.0 - l)
end

MATLAB

function tmp = code(K, m, n, M, l)
	tmp = 1.0 - l;
end

Wolfram

code[K_, m_, n_, M_, l_] := N[(1.0 - l), $MachinePrecision]

Derivation

1. Initial program 74.8%

   \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]
2. Step-by-step derivation

   1. sub-neg 74.8%

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \color{blue}{\left(\ell + \left(-\left|m - n\right|\right)\right)}} \]

   2. associate--r+ 74.8%

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\color{blue}{\left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell\right) - \left(-\left|m - n\right|\right)}} \]

   3. exp-diff 23.6%

      \[\leadsto \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot \color{blue}{\frac{e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell}}{e^{-\left|m - n\right|}}} \]

   4. associate-*r/ 23.6%

      \[\leadsto \color{blue}{\frac{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell}}{e^{-\left|m - n\right|}}} \]

   5. associate-/l* 23.6%

      \[\leadsto \color{blue}{\frac{\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right)}{\frac{e^{-\left|m - n\right|}}{e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell}}}} \]

   6. associate-*r/ 23.6%

      \[\leadsto \frac{\cos \left(\color{blue}{K \cdot \frac{m + n}{2}} - M\right)}{\frac{e^{-\left|m - n\right|}}{e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell}}} \]

   7. exp-diff 17.4%

      \[\leadsto \frac{\cos \left(K \cdot \frac{m + n}{2} - M\right)}{\frac{e^{-\left|m - n\right|}}{\color{blue}{\frac{e^{-{\left(\frac{m + n}{2} - M\right)}^{2}}}{e^{\ell}}}}} \]

3. Simplified 74.8%

   \[\leadsto \color{blue}{\frac{\cos \left(K \cdot \frac{m + n}{2} - M\right)}{e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|n - m\right|\right)}}} \]

4. Taylor expanded in l around inf 31.0%

   \[\leadsto \frac{\cos \left(K \cdot \frac{m + n}{2} - M\right)}{e^{\color{blue}{\ell}}} \]

5. Taylor expanded in K around 0 36.0%

   \[\leadsto \frac{\color{blue}{\cos \left(-M\right)}}{e^{\ell}} \]
6. Step-by-step derivation

   1. cos-neg 36.0%

      \[\leadsto \frac{\color{blue}{\cos M}}{e^{\ell}} \]

7. Simplified 36.0%

   \[\leadsto \frac{\color{blue}{\cos M}}{e^{\ell}} \]

8. Taylor expanded in l around 0 6.5%

   \[\leadsto \color{blue}{-1 \cdot \left(\ell \cdot \cos M\right) + \cos M} \]
9. Step-by-step derivation

   1. +-commutative 6.5%

      \[\leadsto \color{blue}{\cos M + -1 \cdot \left(\ell \cdot \cos M\right)} \]

   2. mul-1-neg 6.5%

      \[\leadsto \cos M + \color{blue}{\left(-\ell \cdot \cos M\right)} \]

   3. unsub-neg 6.5%

      \[\leadsto \color{blue}{\cos M - \ell \cdot \cos M} \]

   4. *-commutative 6.5%

      \[\leadsto \cos M - \color{blue}{\cos M \cdot \ell} \]

10. Simplified 6.5%

    \[\leadsto \color{blue}{\cos M - \cos M \cdot \ell} \]

11. Taylor expanded in M around 0 6.5%

    \[\leadsto \color{blue}{1 - \ell} \]

12. Final simplification 6.5%

    \[\leadsto 1 - \ell \]

Reproduce

herbie shell --seed 2023175 
(FPCore (K m n M l)
  :name "Maksimov and Kolovsky, Equation (32)"
  :precision binary64
  (* (cos (- (/ (* K (+ m n)) 2.0) M)) (exp (- (- (pow (- (/ (+ m n) 2.0) M) 2.0)) (- l (fabs (- m n)))))))